Window function

In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is 'multiplied' by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the 'view through the window'. Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions. w [ n ] = ∑ k = 0 K ( − 1 ) k a k cos ⁡ ( 2 π k n N ) , 0 ≤ n ≤ N . {displaystyle w=sum _{k=0}^{K}(-1)^{k}a_{k};cos left({frac {2pi kn}{N}} ight),quad 0leq nleq N.}     (Eq.1) In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is 'multiplied' by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the 'view through the window'. Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions. The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in each application by requirements like time and frequency resolution. But that method also changes the frequency content of the signal by an effect called spectral leakage. Window functions allow us to distribute the leakage spectrally in different ways, according to the needs of the particular application. There are many choices detailed in this article, but many of the differences are so subtle as to be insignificant in practice. In typical applications, the window functions used are non-negative, smooth, 'bell-shaped' curves. Rectangle, triangle, and other functions can also be used. A rectangular window does not modify the data segment at all. It's only for modelling purposes that we say it multiplies by 1 inside the window and by 0 outside. A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, and, more specifically, that the function goes sufficiently rapidly toward zero. Window functions are used in spectral analysis/modification/resynthesis, the design of finite impulse response filters, as well as beamforming and antenna design. The Fourier transform of the function cos ωt is zero, except at frequency ±ω. However, many other functions and waveforms do not have convenient closed-form transforms. Alternatively, one might be interested in their spectral content only during a certain time period.